Character design consists of different elements, and you should master them all if you want to be a good character designer.
The most essential and basic element is conveying the personality of the character. You need to study how different facial features, facial contours, hairstyles, body shapes, clothing...etc contribute to the overall "vibe" of the character. For example, let's say you have two female characters--one is a young military officer in civilian clothes--she's home visiting her family, the other is the same age--a tomboy from next door who grew up with 3 brothers. Now, both have something about them that makes them different from the average female--something a bit more masculine, but how do you differentiate them in design so we can immediately tell which is the military officer and which is the tomboy? You have to consider their life experiences, what they went through, how their lifestyles dictate how they dress, carry themselves, groom, and affect their facial expressions. And let's say they are both supposed to be attractive, how would you design them so they aren't just typical attractive females, but also convey their backgrounds and personality so we can see something in them that expresses their strength and character? And how would growing up with 3 brothers be different from having gone through military training with a bunch of macho guys? How would all of this be reflected in the visual design of these two characters?
The second element is logic and common sense. You must understand how and why people dress the way they do, why uniforms must function a certain way, or why armors must articulate a certain way. Don't design some sexy chick wearing skimpy outfits if she lives in a very cold area--that's one of the most common mistakes that shallow artists make. Yes, sex sells, but you don't have to be at that level of crassness in order to convey sexiness.
Same thing with hairstyles. Think about what the character does for a living and for hobbies. Would long hair even be practical or safe? Would long hair even be allowed? Would long bangs be distracting for the job/hobby if it constantly falls in front of the person's eyes?
Other things involving long painted nails, wearing high-heels, and various other ridiculous female character designs I've seen that's completely inappropriate for who the character is--they are all part of the common sense/logic problem. All concept artists that put female warrior/soldiers in combat outfits that involves high-heels need to be smacked upside the head.
The third element is aesthetics. I don't mean aesthetics only in attractiveness, but even in how unattractive a character needs to be in order to convey what the character needs to be in the story. If a villain is very attractive, that would be a very different vibe from a villain that is only average looking, like a typical Joe Blow who could vanish into the crowd, or a ugly mofo who would give children nightmares. The understanding of what makes for a beautiful or hideous looking character, and all the range in-between is a very important set of knowledge. There are many professional artists out there who seems to be incapable of depicting exactly the correct range when asked to do so, because they have chosen a very limited range of stylistic vocabulary when they formulated their specific stylization choice, or they simply don't know how to differentiate between minor changes in attractive levels. So if the client asks for a character that's "mildly attractive," would you know how to nail the exact range of attractiveness? What if you are asked to depict a character that's drop dead gorgeous, but there's something slightly awry about the face--something ineffable but noticeable? Would you know how to convey that in the character design?
The fourth element is stylization. From very cartoony to very realistic and all the range in-between. You must consider the tone of the IP you're working on. It is mature? Children-friendly? Horror genre? Fantasy? Sci-fi? Drama? Sexual? How does the IP's genre and target audience influence your stylization choices? It is a very complex subject and not enough people understand enough about it.
If I describe a style that rely on large round eyes, small noses and mouths, round facial contours, idealized cuteness, larger than normal proportioned heads compared to real people, long legs, slender waists, and so on, what would you think of? What if I told you what I described can form drastically different styles using the same set of parameters as I just described? Think about it--Disney and anime/manga both use those same set of stylistic parameters, but they are drastically different--so different that you would never mistake them from each other. So what actually makes those two styles so different if they can be described with the same set of attributes? What about Hanna Barbara? Warner Bros? Don Bluth?
Stylization also involves production concerns. What medium are you designing for? Games? Movies? Feature Animation? TV animation? Comic books? Different mediums have different production requirements, and you must understand them or else you'll run into problems, such as creating unnecessarily complicated characters that cannot be conveyed properly in low-polygon format for a video game. If the IP will generate products in different mediums, will your design translate well across all different mediums?
Stylization also involves knowing your target audience. What culture is the product aimed for? What age range? What's the dominant sex of the target audience? Will the product be localized into other languages and in other countries/cultures? Would your designs have universal appeal across cultures?
And after you combine all these elements, you'll get a general overview that will inform you whether your designs actually fit the IP you're working on.
These are just some of the things you must consider as a character concept artist. There are more, but I think this is already plenty enough for you to chew on for the next few of years of your artistic development.
BTW, I teach all of this stuff in my workshop, "Becoming A Better Artist: Critical Knowledge and Techniques For Today's Artists." You can find out more about it from the link in my signature.